Finite Group Databasebeta

← #46847877

Subgroups — #46847877

Order 34688.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576191C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3Q8×C22⋊D271C211yesmaximal, characteristic, complemented
4#31774544C211yesmaximal, characteristic
5#31774288C211yesmaximal, characteristic, complemented
6#31774141C211yesmaximal, characteristic, complemented
7#31774137C211yesmaximal, characteristic, complemented
8#31774117C211yesmaximal, characteristic, complemented
9#31774101C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774054C211yesmaximal, characteristic, complemented
12#31774046C211yesmaximal, characteristic, complemented
13#31774000C211yesmaximal, characteristic, complemented
14#31773971C211yesmaximal, characteristic, complemented
15#448×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576191—24no—
20C2×Dic1084C2211yescharacteristic, complemented
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864644C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864641C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27SD16×D271—24no—
28C2×#16864638C2211yescharacteristic, complemented
29C2×C8×D271C2211yescharacteristic
30#22576158—21no—
31#22576158C2211yescharacteristic, complemented
32#22576147C2211yescharacteristic
33#22576142—21no—
34#22576140C2211yescharacteristic
35#22576139—21no—
36#22576139C2211yescharacteristic, complemented
37#22576136—21no—
38#22576136C2211yescharacteristic
39#22576131—21no—
40#22576128—21no—
41Q8×Dic271C2211yescharacteristic, complemented
42#22576122C2211yescharacteristic
43C4×C22⋊D271—21no—
44C4×C22⋊D271C2211yescharacteristic
45C22⋊C4×D271—21no—
46#22576093C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic
51#22576081C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic
53#22576070C2211yescharacteristic, complemented
54C22×Q8×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic
56C2×SD16×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60C22⋊C8×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864654—22no—
63#16864654—41no—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—41no—
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271D411yescharacteristic
73D4×D271—22no—
74D4×D271—41no—
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271C2311yescharacteristic
79Dic1084—41no—
80#16864644—22no—
81#16864644—22no—
82#16864643—22no—
83#16864642—22no—
84#16864641—22no—
85#16864641D414yescomplemented
86#16864638—22no—
87C2×C271⋊C8C2311yescharacteristic
88C2×C271⋊C8C2311yescharacteristic
89C8×D271—22no—
90C4⋊Dic271—21no—
91C4⋊Dic271—21no—
92C4⋊Dic271C2311yescharacteristic
93Dic271⋊C4—21no—
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4D411yescharacteristic
98C22⋊Dic271—21no—
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271D411yescharacteristic
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4D411yescharacteristic
106C4×Dic271—21no—
107C4×Dic271C2311yescharacteristic
108C2×Q8×C271—21no—
109C2×Q8×C271—22no—
110C2×Q8×C271—41no—
111C2×Q8×C271C2311yescharacteristic
112C2×Q8×C271C2311yescharacteristic
113C2×D4×C271—21no—
114C2×D4×C271C2311yescharacteristic
115C22×C4×C271—21no—
116C22×C4×C271C2311yescharacteristic
117SD16×C271—22no—
118SD16×C271—41no—
119C2×C8×C271C2311yescharacteristic
120C2×C8×C271C2311yescharacteristic
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271—42no—
125C22×D271—42no—
126C22×D271—81no—
127C22×D271C2×D411yescharacteristic
128Dic542—21no—
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542—42no—
133D1084—21no—
134D1084C2×D412yes—
135C22⋊D271—22no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271C2×D411yescharacteristic
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C4×D271—21no—
145C4×D271—21no—
146C4×D271—42no—
147C4×D271—42no—
148C4×D271—81no—
149C271⋊C8—21no—
150C271⋊C8C2×D412yes—
151C23×C271—21no—
152C23×C271C2×D411yescharacteristic
153Q8×C271—21no—
154Q8×C271—21no—
155Q8×C271—21no—
156Q8×C271—42no—
157Q8×C271C2×D412yes—
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—81no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—42no—
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2411yescharacteristic
169C8×C271—21no—
170C8×C271—21no—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271C22×D411yescharacteristic
178C2×D271—21no—
179C2×D271—21no—
180C2×D271—22no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—84no—
184C2×D271#13712yescomplemented
185C4×C271—21no—
186C4×C271—22no—
187C4×C271—41no—
188C4×C271—41no—
189C4×C271C22×D411yescharacteristic
190C4×C271H(5,𝔽2)11yescharacteristic
191Dic271—21no—
192Dic271—21no—
193Dic271—21no—
194Dic271—41no—
195Dic271—41no—
196Dic271—41no—
197C2×C271—41no—
198C2×C271—81no—
199C2×C271D4211yescharacteristic
200C2×C271C2×#13711yescharacteristic
201C2×C271#57711yescharacteristic
202D271—22no—
203D271—42no—
204C271#326611yescharacteristic, complemented, Sylow
205#3266—2711nomaximal, Sylow
206C2×#137—2711no—
207C22×SD16—2711no—
208D4×Q8—2711no—
209#546—2711no—
210#497—2711no—
211#468—2711no—
212#467—2711no—
213#464—2711no—
214#461—2711no—
215#452—2711no—
216#451—2711no—
217#450—2711no—
218#448D27111yescharacteristic, complemented
219#440—2711no—
220#435—2711no—
221C2×#41—2711no—
222C22×Q8—2711no—
223C22×Q8C2×D27111yescharacteristic, complemented
224C22×D4—2711no—
225#137—5424no—
226C2×Q16—2711no—
227C2×Q16—2711no—
228C2×Q16—2711no—
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16—5424no—
234C2×SD16C2×D27111yescharacteristic
235C2×SD16C2×D27111yescharacteristic
236C2×M16—2711no—
237C22×C8—2711no—
238C4⋊Q8—2711no—
239C4⋊Q8—5421no—
240#124—2711no—
241#123—5421no—
242C22⋊Q8—2711no—
243C22⋊Q8—2711no—
244C22⋊Q8—2711no—
245C22⋊Q8—5421no—
246C22⋊Q8—5421no—
247#121—5421no—
248#121C2×D27111yescharacteristic
249#120—5421no—
250C4×Q8—2711no—
251C4×D4—2711no—
252C4×D4—2711no—
253C4×D4—5421no—
254C2×C22⋊C4—5421no—
255#107—2711no—
256C4⋊C8—2711no—
257#103—2711no—
258#103—2711no—
259#103C2×D27111yescharacteristic, complemented
260#103C2×D27111yescharacteristic, complemented
261#102—2711no—
262#102—2711no—
263C22⋊C8—2711no—
264C22⋊C8C2×D27111yescharacteristic, complemented
265#97—2711no—
266C24—5421no—
267#41—5422no—
268#41—10841no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2711no—
273C2×Q8—2711no—
274C2×Q8—5421no—
275C2×Q8—5422no—
276C2×Q8—10841no—
277C2×Q8C22×D27111yescharacteristic
278C2×Q8C22×D27111yescharacteristic
279C2×D4—21no—
280C2×D4—2711no—
281C2×D4—2711no—
282C2×D4—5421no—
283C2×D4—5422no—
284C2×D4—5421no—
285C2×D4—10841no—
286C2×D4C22×D27111yescharacteristic
287C22×C4—21no—
288C22×C4—2711no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4—5421no—
292C22×C4C22×D27111yescharacteristic
293Q16—5422no—
294Q16—5422no—
295Q16—10841no—
296SD16—22no—
297SD16—41no—
298SD16—2714no—
299SD16—5422no—
300SD16—5422no—
301SD16—5422no—
302M16—5422no—
303C2×C8—2711no—
304C2×C8—2711no—
305C2×C8—5422no—
306C2×C8C22×D27111yescharacteristic
307C2×C8C22×D27111yescharacteristic
308C4⋊C4—2711no—
309C4⋊C4—2711no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4—5421no—
313C4⋊C4—5421no—
314C4⋊C4—5421no—
315C4⋊C4—5421no—
316C4⋊C4C22×D27111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2711no—
319C22⋊C4—2711no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C42—2711no—
327C42—5421no—
328C23—21no—
329C23—2711no—
330C23—5421no—
331C23—10842no—
332C23—10842no—
333C23—21681no—
334C23D4×D27111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—5421no—
340Q8—5421no—
341Q8—5421no—
342Q8—5421no—
343Q8—10842no—
344Q8D4×D27112yes—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—2712no—
349D4—5422no—
350D4—5421no—
351D4—10842no—
352D4—10842no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—2711no—
360C2×C4—2711no—
361C2×C4—2711no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—5421no—
366C2×C4—5421no—
367C2×C4—10842no—
368C2×C4—10842no—
369C2×C4—21681no—
370C2×C4D4×D27111yescharacteristic
371C2×C4C23×D27111yescharacteristic
372C8—21no—
373C8—21no—
374C8—2712no—
375C8—5421no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—2712no—
383C22—5421no—
384C22—5421no—
385C22—5422no—
386C22—10841no—
387C22—10841no—
388C22—21684no—
389C22C2×D4×D27111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—5421no—
395C4—5421no—
396C4—5421no—
397C4—10841no—
398C4—10841no—
399C4—10841no—
400C4C2×D4×D27111yescharacteristic
401C4#2257620511yescharacteristic
402C2—41no—
403C2—81no—
404C2—5422no—
405C2—10842no—
406C2#3177452911yescharacteristic
407C2#137×D27111yescharacteristic
408C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#448×C271C2
socleC22×C271C22×D4
prime radical#31774544C2
hypercenter#448D271
hypocenterC271#3266
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22576191C2semidirect
2. C2×SD16×D271C2semidirect
3. Q8×C22⋊D271C2semidirect
5. #31774288C2semidirect
6. #31774141C2semidirect
7. #31774137C2semidirect
8. #31774117C2semidirect
9. #31774101C2semidirect
10. #31774061C2semidirect
11. #31774054C2semidirect
12. #31774046C2semidirect
13. #31774000C2semidirect
14. #31773971C2semidirect
15. #448×C271C2semidirect
20. C2×Dic1084C22semidirect
21. C2×#16864644C22semidirect
22. C2×#16864644C22semidirect
28. C2×#16864638C22semidirect
31. #22576158C22semidirect
36. #22576139C22semidirect
41. Q8×Dic271C22semidirect
46. #22576093C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
49. #22576083C22semidirect
53. #22576070C22semidirect
54. C22×Q8×C271C22semidirect
58. #103×C271C22semidirect
59. #103×C271C22semidirect
60. C22⋊C8×C271C22semidirect
85. #16864641D4semidirect
184. C2×D271#137semidirect
204. C271#3266semidirect
218. #448D271semidirect
223. C22×Q8C2×D271semidirect
259. #103C2×D271semidirect
260. #103C2×D271semidirect
264. C22⋊C8C2×D271semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225761912
C2×SD16×D2712
Q8×C22⋊D2712
#317745442
#317742882
#317741412
#317741372
#317741172
#317741012
#317740612
#317740542
#317740462
#317740002
#317739712
#448×C2712
#3266271